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Simple models of neurons in rodent brains

Overview

This chapter compares the RTM excitatory-neuron model with Wang--Buzsáki (WB) and Erisir inhibitory-neuron models. All are conductance-based, but their maximal conductances, reversal potentials, and gating kinetics produce distinct spike waveforms and recovery speeds.

Core ideas

The models share sodium, potassium, and leak currents, while their rate laws encode cell-type-specific kinetics. In these implementations, sodium activation $m$ is set instantaneously to its steady-state value; $h$ and $n$ remain dynamic. The two Erisir traces differ in the power used for the potassium gate, making that modeling choice directly visible.

Essential model

The common current-balance form is

$$ C\frac{dV}{dt}=I_{\mathrm{ext}}-g_{\mathrm{Na}}m_\infty(V)^3h(V-E_{\mathrm{Na}}) -g_{\mathrm{K}}n^p(V-E_{\mathrm{K}})-g_{\mathrm{L}}(V-E_{\mathrm{L}}). $$

Here $V$ is membrane voltage, $t$ is time, $C$ is capacitance, $I_{\mathrm{ext}}$ is applied current, $g_{\mathrm{Na}}$, $g_{\mathrm{K}}$, and $g_{\mathrm{L}}$ are maximal conductances, $E_{\mathrm{Na}}$, $E_{\mathrm{K}}$, and $E_{\mathrm{L}}$ are reversal potentials, $m_\infty$ is the instantaneous sodium-activation value, $h$ is sodium inactivation, $n$ is potassium activation, and $p$ is the potassium-gate exponent selected by the model. Dynamic gates obey $dx/dt=\alpha_x(V)(1-x)-\beta_x(V)x$, where $x$ is a gate and $\alpha_x$, $\beta_x$ are its voltage-dependent rates.

Code examples

All five examples now live in one notebook, chapter05.ipynb: simulate_rtm_voltage_trace runs the RTM model; simulate_rtm_gating_variables plots RTM gate steady states and time constants; simulate_wb_voltage_trace and simulate_wb_voltage_trace_1996 run the two WB parameterizations; simulate_erisir_voltage_trace(n_power=...) covers both Erisir variants ($n^2$ and $n^4$). Each has an ipywidgets slider to explore its parameters interactively.

What to look for

Compare voltage traces before treating a label such as “excitatory” or “inhibitory” as a complete explanation. In the gating figure, compare both the equilibrium curves and their time constants; rate laws, not merely the names of the gates, distinguish the models.

Suggested order

  1. Begin with THREE_MODELS_GATING_VARIABLES and the RTM rate functions.
  2. Run RTM_VOLTAGE_TRACE, then WB_VOLTAGE_TRACE.
  3. Compare the two Erisir variants and locate the changed power of $n$.

Prerequisites and related chapters

Chapters 01 and 03 introduce conductance-based HH-style models and gate kinetics. Chapter 09 extends the RTM model with slow adaptation currents.

Running the examples

Open chapter05.ipynb in Jupyter, or via the Colab badge at the top of the notebook. Run all cells top to bottom; each section's static figure reproduces the book's plot, and the interact(...) cell below it lets you adjust that example's parameters with sliders.